Exactness in the Path Integral of the Coulomb Potential in One Space Dimension

TL;DR

This paper provides an exact solution to the time-sliced path integrals of the one-dimensional Coulomb system using advanced transformation and gauge techniques, clarifying the path integral's exactness through pole analysis.

Contribution

It introduces an exact formulation of the path integral for the 1D Coulomb potential using the Duru-Kleinert transformation and Fujikawa's gauge method, enabling precise evaluation.

Findings

Exact Feynman kernels for bound and scattering states obtained

Pole structure explains the path integral's exactness

Path integrals can be evaluated exactly via Cauchy's theorem

Abstract

We solve time-sliced path integrals of one-dimensional Coulomb system in an exact manner. In formulating path integrals, we make use of the Duru-Kleinert transformation with Fujikawa's gauge theoretical technique. Feynman kernels in the momentum representation both for bound states and scattering states will be obtained with clear pole structure that explains the exactness of the path integral. The path integrals presented here can be, therefore, evaluated exactly by making use of Cauchy's integral theorem.